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Existence and nonexistence of hypercyclic semigroups

Authors: L. Bernal-González and K.-G. Grosse-Erdmann
Journal: Proc. Amer. Math. Soc. 135 (2007), 755-766
MSC (2000): Primary 47A16; Secondary 47D03
Published electronically: August 31, 2006
MathSciNet review: 2262871
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Abstract: In these notes we provide a new proof of the existence of a hypercyclic uniformly continuous semigroup of operators on any separable infinite-dimensional Banach space that is very different from--and considerably shorter than--the one recently given by Bermúdez, Bonilla and Martinón. We also show the existence of a strongly dense family of topologically mixing operators on every separable infinite-dimensional Fréchet space. This complements recent results due to Bès and Chan. Moreover, we discuss the Hypercyclicity Criterion for semigroups and we give an example of a separable infinite-dimensional locally convex space which supports no supercyclic strongly continuous semigroup of operators.

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Additional Information

L. Bernal-González
Affiliation: Departamento de Análisis Matemático, Facultad de Matemáticas, Apdo. 1160, Avda. Reina Mercedes, 41080 Sevilla, Spain

K.-G. Grosse-Erdmann
Affiliation: Fachbereich Mathematik, Fernuniversität Hagen, 58084 Hagen, Germany

Keywords: Hypercyclic uniformly continuous semigroup of operators, topologically mixing semigroup, Hypercyclicity Criterion, supercyclic semigroup
Received by editor(s): November 29, 2004
Received by editor(s) in revised form: October 5, 2005
Published electronically: August 31, 2006
Additional Notes: The first author was partially supported by Plan Andaluz de Investigación de la Junta de Andalucía FQM-127 and by Ministerio de Ciencia y Tecnología Grant BFM2003-03893-C02-01
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.